Symmetry and transitive properties of monohedral f-triangulations of the Riemannian sphere

Here we give the complete description of the symmetry group and transitive properties of the set of all of monohedral triangulations of the Riemannian sphere by f-tilings. We shall also show that each monohedral f-tiling of the Riemannian sphere can be seen, up to a spherical isometry, as the singular set of a spherical isometric folding.